Polygonal Quasiconformality and Grunsky’s Operator

This paper concerns the old problem of the connection between the dilatations of a given quasisymmetric homeomorphism h of a circle and the associated polygonal quasiconformal maps with a fixed finite number of boundary points, namely whether k(h)=supkn, where the supremum is taken over all possible n-gons formed by the disk with n distinguished boundary points. A still open question is whether such equality is valid under the additional assumption that the naturally related univalent functions with quasiconformal extensions have equal Grunsky and Teichmüller norms. We solved this problem in the negative for n≥4.